Classical electrodynamics phenomena in QED

[nightvid08]

I find the usual formalism of QED very ad hoc, opaque and hard to follow - and one particular thing has been bugging me for quite some time. Consider the following 3 QED processes:

1. Coulomb scattering of an electron or a positron from a heavy nucleus
2. Bremsstrahlung radiation during Coulomb scattering
3. Thomson/Compton scattering

Classically, several things, which could easily be shown in Jackson-level classical electrodynamics, seem intuitively obvious:

1. The differential cross-section is modified if an electron moves relativistically when near the nucleus, even if it has VERY low energy far away (say, 10 eV). This would not happen for positrons.
2. Electrons emit far more bremsstrahlung at low energy than positrons because (classically speaking) their acceleration is more violent as they pass near the nucleus. Also, relativistic effects can modify the bremsstrahlung for electrons in this case, even if the energy far away is very low, while this does not happen for positrons.
3. In Thomson scattering, the cross-section is modified if the
e.m. radiation is of such a high intensity that the electron oscillates at a relativistic speed during the scattering process. Presumably, the cross-section for Compton scattering should also be modified if the "classical oscillation velocity" of the electron is relativistic.

All of these effects, which make so much sense classically, have been confirmed by experiment. And yet it is far from obvious how to reproduce the CED results in QED...

Is anyone else bothered by the fact that most QFT books leave you completely baffled as to how to understand any of it?